Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 28-p, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10314, whose string of implications is:
    0 \(\Rightarrow\) 0
  • A proven non-implication whose code is 3. In this case, it's Code 3: 82, Form 0 \( \not \Rightarrow \) Form 182 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 182 <p> There is an aleph whose cofinality is greater than \(\aleph_{0}\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 5514, whose string of implications is:
    28-p \(\Rightarrow\) 427 \(\Rightarrow\) 67 \(\Rightarrow\) 89 \(\Rightarrow\) 90 \(\Rightarrow\) 51 \(\Rightarrow\) 25 \(\Rightarrow\) 34 \(\Rightarrow\) 104 \(\Rightarrow\) 182

The conclusion Form 0 \( \not \Rightarrow \) Form 28-p then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M17\) Gitik's Model Using the assumption that for every ordinal \(\alpha\) there is a strongly compact cardinal \(\kappa\) such that \(\kappa >\alpha\), Gitik extends the universe \(V\) by a filter \(G\) generic over a proper class of forcing conditions

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